Cruz, who is running for president, tried to get unanimous consent to pass his legislation — referred to as "Kate's Law," after Kathryn Steinle, who was shot and killed in San Francisco allegedly by an illegal immigrant who had already been deported five times.

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Under the proposal, undocumented immigrants would face additional prison time if they reenter the country, including a minimum five-year sentence if they were previously convicted of an aggravated felony or of illegally reentering the country twice.

Cruz said Congress needs "leadership" on the issue, adding, "enough hot air. ... With whom do you stand? Do you stand with violent criminal illegal aliens or do you stand with American citizens?"

"The new mandatory minimum sentences created by this bill would have a crippling financial effect, and that's an understatement, with no evidence that they would deter future violations of the law," he added.

Reid added on Wednesday that since the Senate passed immigration reform in 2013, "All we've seen from Republican leaders, and their caucuses, is bills attacking immigrants."

The Texas Republican fired back, saying it was "sad" that Reid "chooses to stand with violent illegal criminal aliens instead of the American citizens" and is boxing legal immigrants together with undocumented immigrants.

"I am the son of an immigrant who came legally from Cuba," he added. "For the Democratic leader to cynically suggest that somehow immigrants should be lumped into the same bucket with murderers and rapists demonstrates the cynicism of the modern Democratic Party."